Arkham

Lady is a demon hunter. The reason she is climbing the tower is so she can stop her father, Arkham, from unleashing the demon world. With her trusty rocket launcher she thinks she can take him down. She then meets Dante and tries to kill him. She shoots a rocket and Dante expertly rides it. She meets her father half-way through the game and thinks she killed him, but he comes back. Also he tells her Vergil possesed him which isn't true. At mission 17 Dante fights her again. After defeating her she gives him her rocket launcher.